Rabbi Abadi’s distinction is between cheeses that have some kind of rennet (even non-animal rennet) and those that do not. Cheeses that require rennet were subject to the Gezeira and cheeses that do not require rennet were not subject to the Gezeira.
Rav Moshe holds like this as well and I believe (but would like confirmation) this is the shita followed by the OU.
I believe Rabbi Abadi used to be machmir and held that all cheeses were subject to the gezeira of gevinas akum and he only recently (past 8 years or so) changed his mind and to hold like Rav Moshe.